These, then, are characterizations of the system of natural numbers within an ambient set theory. And they seem to work, in the sense that in a sufficiently strong set theory it can be shown that Peano's axioms have (up to isomorphism) a unique model (cf. Rem. 6.1.8).

2008, Akihiro Kanamori, The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings, Springer Science & Business Media →ISBN, page 369

As much of the work in determinacy must proceed without AC, ZF serves as the ambient theory for this section, and uses of AC will be explicitly noted, reversing the usual procedure.